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SpatialSTEM:
A Mathematical/Statistical Framework for Understanding and Communicating
Grid-based Map Analysis and Modeling
Premise:
There
a “map-ematics” that
that extends
math/stat
concepts
Premise:
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extendstraditional
traditional
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the
quantitative analysis
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variables
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This presentation provides a fresh perspective on interdisciplinary instruction
at the college level by combining the philosophy and approach of STEM with the
spatial reasoning and analytical power of grid-based Map Analysis and Modeling
This PowerPoint with notes and online links to further reading is posted at
www.innovativegis.com/basis/Courses/SpatialSTEM/
Presented by
Joseph K. Berry
Adjunct Faculty in Geosciences, Department of Geography, University of Denver
Adjunct Faculty in Natural Resources, Warner College of Natural Resources, Colorado State University
Principal, Berry & Associates // Spatial Information Systems
Email: [email protected] — Website: www.innovativegis.com/basis
Geotechnology
(Nanotechnology)
(Biotechnology)
Geotechnology is one of the three "mega technologies" for the 21st century and
promises to forever change how we conceptualize, utilize and visualize
spatial relationships in scientific research and commercial applications (U.S. Department of Labor)
Geographic Information
Systems (map and analyze)
Remote Sensing
Global Positioning
System (location and navigation)
(measure and classify)
GPS/GIS/RS
The Spatial Triad
Computer Mapping (70s) — Spatial Database Management (80s) — Map Analysis (90s) — Multimedia Mapping (00s)
is
Technological Tool
Analytical Tool
Where
What
Mapping
involves precise
placement
(delineation) of
physical features
(graphical inventory)
Modeling involves
Descriptive
Mapping
Why
Prescriptive
Modeling
So What
and
What If
analysis of spatial
patterns and
relationships
(map analysis/modeling)
(Berry)
A Mathematical Structure for Map Analysis/Modeling
Geotechnology
Technological Tool
Mapping/Geo-Query
(Discrete, Spatial Objects)
 RS – GIS – GPS
(Continuous, Map Surfaces)
Analytical Tool
Map Analysis/Modeling
Geo-registered
Map Stack
“Map-ematics”
Analysis Frame  Matrix
of Numbers
Maps as Data, not Pictures
Vector & Raster — Aggregated & Disaggregated
Qualitative & Quantitative
…organized set of numbers
Grid-based
Map Analysis
Spatial Analysis Operations
Spatial Statistics Operations
Toolbox
GISer’s Perspective:
Reclassify and Overlay
Distance and Neighbors
Local
Functions
Focal
Surface Modeling
Spatial Data Mining
Zonal
Global
Operators
Procedures
Mathematician’s Perspective:
Basic GridMath & Map Algebra
Advanced GridMath
Map Calculus
Map Geometry
Plane Geometry Connectivity
Solid Geometry Connectivity
Unique Map Analytics
GISer’s Perspective:
Statistician’s Perspective:
The SpatialSTEM
Framework
Traditional math/stat procedures
can be extended into
geographic space to stimulate
those with diverse backgrounds
and interests for…
“thinking analytically
with maps”
Basic Descriptive Statistics
Basic Classification
Map Comparison
Unique Map Statistics
Surface Modeling
Advanced Classification
Predictive Statistics (Berry)
Spatial Analysis Operations (Geographic Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Grid Layer
Map Stack
Spatial Analysis
extends the basic set of discrete map features (points, lines and polygons) to
map surfaces that represent continuous geographic space as a set of contiguous grid cells (matrix),
thereby providing a Mathematical Framework for map analysis and modeling of the
Contextual Spatial Relationships within and among grid map layers
Map Analysis Toolbox
Unique spatial
operations
Mathematical Perspective:
Basic GridMath & Map Algebra ( + - * / )
Advanced GridMath (Math, Trig, Logical Functions)
Map Calculus (Spatial Derivative, Spatial Integral)
Map Geometry (Euclidian Proximity, Effective Proximity, Narrowness)
Plane Geometry Connectivity (Optimal Path, Optimal Path Density)
Solid Geometry Connectivity (Viewshed, Visual Exposure)
Unique Map Analytics (Contiguity, Size/Shape/Integrity, Masking, Profile)
(Berry)
Spatial Analysis Operations (Math Examples)
Advanced Grid Math — Math, Trig, Logical Functions
Map Calculus — Spatial Derivative, Spatial Integral
Spatial Derivative
MapSurface
2500’
…is equivalent to the slope
of the tangent plane at a
location
Slope draped over
MapSurface
500’
Surface
Fitted Plane
65%
SLOPE MapSurface Fitted
FOR MapSurface_slope
0%
Curve
The derivative is the
instantaneous “rate of
change” of a function and
is equivalent to the slope
of the tangent line at
a point
Dzxy Elevation
ʃ Districts_Average Elevation
Spatial Integral
Advanced Grid Math
…summarizes the values on a
surface for specified map areas
(Total= volume under the surface)
Surface Area
S_Area=
Fn(Slope)
…increases with
increasing inclination
as a Trig function of
the cosine of
the slope
angle
COMPOSITE Districts WITH MapSurface
Average FOR MapSurface_Davg
MapSurface_Davg
S_area= cellsize / cos(Dzxy Elevation)
The integral calculates the
area under the curve for any
section of a function.
Surface
Curve
(Berry)
Spatial Analysis Operations (Distance Examples)
96.0 minutes
Map Geometry — (Euclidian Proximity, Effective Proximity, Narrowness)
Plane Geometry Connectivity — (Optimal Path, Optimal Path Density)
Solid Geometry Connectivity — (Viewshed, Visual Exposure)
Distance
Euclidean Proximity
…farthest away by truck,
ATV and hiking
Effective Proximity
Off Road
Relative Barriers
HQ (start)
On Road
26.5 minutes
Off Road
Absolute Barrier
…farthest away
by truck
On + Off Road
Travel-Time
Surface
Farthest
(end)
Shortest straight line
between two points…
…from a point to
everywhere…
…not necessarily straight
lines (movement)
Connectivity
HQ
Truck = 18.8 min
ATV = 14.8 min
Hiking = 62.4 min
(start)
…like a raindrop, the
“steepest downhill
path” identifies the
optimal route
Solid Geometry Connectivity
Rise
Run
Plane Geometry
Visual Exposure
(Quickest Path)
Tan = Rise/Run
Seen if new tangent exceeds
all previous tangents
along the line of sight
 Counts
# Viewers
Sums
Viewer
Weights 
Splash
270/621= 43% of the entire
Viewshed
road network is connected
Highest
Weighted
Exposure
(Berry)
Spatial Statistics Operations (Numeric Context)
GIS as “Technical Tool” (Where is What) vs. “Analytical Tool” (Why, So What and What if)
Grid Layer
Map Stack
Spatial Statistics seeks to map the variation in a data set instead of focusing on
a single typical response (central tendency), thereby providing
a Statistical Framework for map analysis and modeling of the
Numerical Spatial Relationships within and among grid map layers
Map Analysis Toolbox
Unique spatial
operations
(Berry)
Statistical Perspective:
Basic Descriptive Statistics (Min, Max, Median, Mean, StDev, etc.)
Basic Classification (Reclassify, Contouring, Normalization)
Map Comparison (Joint Coincidence, Statistical Tests)
Unique Map Statistics (Roving Window and Regional Summaries)
Surface Modeling (Density Analysis, Spatial Interpolation)
Advanced Classification (Map Similarity, Maximum Likelihood, Clustering)
Predictive Statistics (Map Correlation/Regression, Data Mining Engines)
Spatial Statistics (Linking Data Space with Geographic Space)
Roving Window (weighted average)
Geo-registered Sample Data
Spatial Distribution
Spatial
Statistics
Discrete Sample Map
Non-Spatial Statistics
Continuous Map Surface
Surface Modeling techniques are used to derive a continuous map surface
from discrete point data– fits a Surface to the data (maps the variation).
Standard Normal Curve
Average = 22.6
In Geographic Space, the typical value
forms a horizontal plane implying
the average is everywhere to
form a horizontal plane
StDev =
26.2
Histogram
(48.8)
10
20
30
40
50
Numeric Distribution
(Berry)
X + 1StDev
= 22.6 + 26.2
=
In Data Space, a
standard normal curve can
be fitted to the data to identify
the “typical value” (average)
0
…lots of NE locations
exceed Mean + 1Stdev
60
70
80
Unusually
high
values
X= 22.6
+StDev
Average
48.8
Spatial Statistics Operations (Data Mining Examples)
Map Clustering:
Elevation vs. Slope Scatterplot
“data pair”
of map values
“data pair”
plots here in…
Cluster 2
Data
Space
…as similar as can be WITHIN
a cluster …and as different as
can be BETWEEN clusters
Elevation
Geographic
Space
(Feet)
Slope
+
Slope
(Percent)
Slope draped
on Elevation
Elev
X axis = Elevation (0-100 Normalized)
Y axis = Slope (0-100 Normalized)
Advanced Classification (Clustering)
Map Correlation:
+
Cluster 1
Geographic Space
Data Space
Spatially Aggregated Correlation
Scalar Value – one value represents the overall non-spatial relationship
between the two map surfaces
Roving Window
…1 large data table
Entire Map
Extent
Elevation
(Feet)
Slope
(Percent)
with 25rows x 25 columns =
625 map values for map wide summary
r=
…where x = Elevation value and y = Slope value
and n = number of value pairs
…625 small data tables
within 5 cell reach =
81map values for localized summary
Localized Correlation
Predictive Statistics (Correlation)
(Berry)
Map Variable – continuous quantitative
surface represents the localized spatial
relationship between the two map surfaces
r = .432 Aggregated
Map of the Correlation
So What’s the Point? (4 key points)
1) Current
GIS education for the most part insists that non-GIS students interested in
understanding map analysis and modeling must be tracked into general GIS courses
that are designed for GIS specialists, and material presented primarily focus on
commercial GIS software mechanics that GIS-specialists need to know to function
in the workplace.
solutions to complex spatial problems need to engage “domain expertise”
through GIS– outreach to other disciplines to establish spatial reasoning skills needed
for effective solutions that integrate a multitude of disciplinary and general
2) However,
public perspectives.
map analysis and modeling involving Spatial Analysis and Spatial
Statistics are in large part simply spatial extensions of traditional mathematical
and statistical concepts and procedures.
3) Grid-based
recognition by the GIS community that quantitative analysis of maps is a
reality and the recognition by the STEM community that spatial relationships exist
and are quantifiable should be the glue that binds the two perspectives– a common
coherent and comprehensive SpatialSTEM approach.
4) The
Bottom Line
“…map-ematics  quantitative analysis of mapped data”
— not your grandfather’s map …nor his math/stat
Online Book Chapter on the SpatialSTEM Approach
Online book chapter…